We will determine here instead the parameter space that solves both the strong CP problem and the nature of DM within the \(Z_{\cal N}\)  framework under discussion. The question of DM is of strong experimental interest and very timely,
in particular given the plethora of projects targeting light DM candidates, down to the region of “fuzzy” DM [1]}. For instance,
CASPEr-Electric [2]}, [3]}, [4]} probes directly the anomalous gluonic coupling of the axion via an oscillating neutron electric dipole moment (nEDM): the strength of that coupling cannot be modified for the canonical QCD axion irrespective of the model details, unlike other axion-to-SM couplings that can be selectively enhanced [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}.
We show here that, in contrast, a hypothetical early discovery at
CASPEr-Electric could be interpreted as a solution to the strong CP problem via a \(Z_{\cal N}\)  reduced-mass axion, because of the same-size enhancement of all axion couplings
in such a scenario.
